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There is a wide range of applications where the local extrema of a functionare the key quantity of interest. However, there is surprisingly little work onmethods to infer local extrema with uncertainty quantification in the presenceof noise. By viewing the function as an infinite-dimensional nuisanceparameter, a semiparametric formulation of this problem poses dauntingchallenges, both methodologically and theoretically, as (i) the number of localextrema may be unknown, and (ii) the induced shape constraints associated withlocal extrema are highly irregular. In this article, we build upon aderivative-constrained Gaussian process prior recently proposed by Yu et al.(2023) to derive what we call an encompassing approach that indexes possiblymultiple local extrema by a single parameter. We provide closed-formcharacterization of the posterior distribution and study its large samplebehavior under this unconventional encompassing regime. We show that theposterior measure converges to a mixture of Gaussians with the number ofcomponents matching the underlying truth, leading to posterior exploration thataccounts for multi-modality. Point and interval estimates of local extrema withfrequentist properties are also provided. The encompassing approach leads to aremarkably simple, fast semiparametric approach for inference on local extrema.We illustrate the method through simulations and a real data application toevent-related potential analysis.

Semiparametric Bayesian Inference for Local Extrema of Functions in the Presence of Noise